Vladimir A. Testov

At present, the task of intellectual development of students comes to the forefront in teaching mathematics at school, and the development of their mathematical thinking is becoming increasingly relevant. The aim of the article is to examine those elements in the procedural-cognitive side of the content of teaching mathematics, which act as mental means and methods of mathematical knowledge. The methodological basis of the study is the systemic, meta-subject and activity-based approaches. As a result of learning mathematics, the mind of a person forms structures corresponding to the identified elements of the content of teaching mathematics, which are called mathematical thinking schemes in the article. These include logical, algorithmic, combinatorial and figurative-geometric cognitive structures. The article provides their characteristics and describes the practical experience of their formation in schoolchildren. All these structures are universal, i.e. they are used regardless of the specific mathematical material, and they are of great importance not only for learning, but also for mathematical creativity. All the considered schemes of mathematical thinking have one common characteristic: their formation can be carried out only over a long period of time, using the sensitive possibilities of their development in each age period. A theoretical analysis of the relationship between the processes of differentiation and integration of such structures is made. It is shown that the process of integration should prevail quantitatively in the formation of such structures, gradual, small-step accumulation of mathematical knowledge. However, qualitative leaps and breakthroughs in the creation of knowledge can occur through differentiation, using the deductive method. The recommendations given in the article to teachers on the formation of various types of mathematical thinking schemes in schoolchildren at different levels of education are of practical significance. The greatest attention is paid to figurative-geometric thinking schemes and their role in teaching mathematics. It is noted that the use of computer mathematics systems in teaching, which also help in conducting computer experiments and in research-based learning, is a great help to the teacher in forming such patterns of thinking in the conditions of a digital society.

Currently, there are many problems in the teaching of mathematics associated with a decrease in students' motivation, many of them do not understand the studied material properly. One of the main reasons for this situation is the reliance in teaching not on the natural features of the perception of mathematical knowledge by children of different ages, but most often on a purely logical sequence of the material. Soviet teaching methods, for ideological reasons, limited direction of attention toward the natural characteristics of students. This led to the fact that a high theoretical level of teaching mathematics at school was combined with the complication of educational programs, absolutization of theoretical thinking in comparison with figurative thinking, and neglect of visual learning tools. The purpose of the article is the development of new approaches to the methodology of teaching mathematics, based on the principle of nature conformity - one of the most famous pedagogical principles. This principle considers the attitude towards a person during training as a part of nature, provides for the reliance on his/her own talents and inclinations given to him/her from birth. For a subject such as mathematics, which is studied from the 1st to the 11th grade, and then also in higher school, the principle of nature-conformity is implemented primarily through the phasing, multi-stage gain of new mathematical knowledge by students. The necessity of preliminary stages, steps in the study of basic mathematical concepts is shown. The phasing of knowledge is examined on the example of the formation of the most important algebraic concept of a group and such a structure as scalar. These two concepts permeate the entire course of school mathematics and some university courses. A new approach to the study of the principle and method of mathematical induction is also considered on the basis of a minimum condition that is clearer for students to understand. Examples of applying this form of induction to solving tasks are given.

The paper discusses the strategy of learning based on social-cultural role of mathematics in education. In the learning process, students must understand how to relate concepts studied by them with the urgent tasks of the practice. Tasks with practical content, mathematical model of which is the inequality or system of inequalities, are given as the example.

The paper discusses the influence of use of socio-cultural experiences to motivate learners to study mathematics. The author distinguishes such components of socio-cultural experience as a genetic approach, elements of historicism, practical and humanitarian orientation, educational and aesthetic aspects of teaching mathematics, idea of mathematics as the language of science.

The paper discusses the peculiarities of pupils’ mathematical notions the formation in the modern paradigm of education and in the light of the demands, made in the concept of mathematical education. These requirements imply updating the content of teaching mathematics at school, bringing it closer to the modern sections and practical applications, the widespread using of project activities. To overcome the existing fragmentation of various mathematical disciplines and the isolation of individual sections, to ensure the integrity and unity in the teaching of mathematics is possible only on by allocating the main lines in it. Mathematical structures are the rods, the main construction lines of mathematical courses. Phased process of formation of concepts about the basic mathematical structures is a prerequisite for the implementation of the principle of availability of training. Method of projects can be of great help in a phased study of mathematical structures. Application of this method in the study of mathematical structures allows solve a number of tasks to expand and deepen the knowledge of mathematics, consider the possibilities of their applica-tion in practice, the acquisition of practical skills to work with modern software products, the full development of the individual abilities of pupils.